\sqrt{53+20\sqrt{7}} can be written in the form a+b\sqrt{c}, where a, b, and c are integers and c has no factors which is a perfect square of any positive integer other than 1. Find a+b+c.

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Explanation:

Since the range of is , this means that cannot be negative.

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Let's set the given nested radical expression equal to the form a+b*sqrt(c), and then square both sides to see what happens

Equating the corresponding parts shows that

Therefore c = 7 must be the case to allow the radicands to match up.

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Since c = 7, we can update the first equation like so

We'll come back to this later.

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Focus on

Note how 20 and 2ab are coefficients outside the root. So 20 = 2ab leads to ab = 10.

Here are all the ways to multiply to 10 where a,b are integers

So we could have these possibilities.

If you try all of those a,b combinations with , then you should find that a = 5 and b = 2 works. I'm skipping a bunch of steps here. All you basically do is plug and chug.

Due to symmetry, we could have a = -5 and b = -2 as another solution to . However, if a < 0 and b < 0, then would be negative which contradicts the fact that is positive. So we'll ignore any case where a,b are negative.

Therefore, the only combination that works is a = 5 and b = 2

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Overall, we found that

And therefore,

You can confirm this with your calculator by taking advantage of the idea "If x = y, then x-y = 0". In other words, if you subtract the left and right sides, you should get 0 as a result (the order of subtraction doesn't matter in this case). Don't forget to use parenthesis correctly, or you may not get 0. It might help to think of it as the form (x) - (y) = 0.

After we confirmed we have the correct a,b,c values, we simply add them up to get a+b+c = 5+2+7 = 14.

The growth factor is A. 0.28 per decade.

Step-by-step explanation:

Step 1; It is given that the city grows by 28% per decade. Now we assign values to get a better understanding. If there were 1,000 people at the start of the decade there would be 1,000 + 28% at the end of the decade = 1000 + 280 = 1,280 at the end of the decade.

Step 2; To calculate the growth factor, we calculate the difference between the present value and the past value and divide it by the past value.

Growth factor = = = . So the growth factor is 0.28 per decade.